Lore

Amygdala is a Great One Boss in Bloodborne, and is found in the Nightmare Frontier. It's similar to the large enemies who attempt to grab the Hunter with portals, but lighter in color, and lacks the tentacles on its head.

From Wikipedia: "The amygdalae (singular: amygdala) are two almond-shaped groups of nuclei located deep and medially within the temporal lobes of the brain in complex vertebrates, including humans. Shown in research to perform a primary role in the processing of memory, decision-making, and emotional reactions, the amygdalae are considered part of the limbic system."

Notes:
1. Amygdala (the boss) has an arm on the right side of its body that is always raised during the boss fight. In the palm of this arm's hand is a glowing eye shape, similar to the one on its chest area.
2. The Amygdala fought in the Nightmare Frontier does not have tentacles on its head like the amygdalae seen in Yharnam.
3. The eyes in Amygdala's head are not always visible.
4. The heads and arms of the Amygdala resemble the Garden of Eyes enemy, especially when the Amygdala eyes are visible.

Theories

Amygdala Means Tonsil

In Greek, the name of this beast translates to tonsil in English. The tonsils basically act as an immune defence storage to prevent infections from the things you eat and breathe. They also look like the head of the Amygdala creature in the game. So, this boss, and similar creatures act as tonsils; they exist to protect the nightmare from infections (like the Hunter).

Dialogue

NB. this may actually be Patches talking, but it doesn't seem to sound like him…After transporting the player to Lecture Theatre F1:
[cackling] "Oh Amygdala, oh Amygdala…
Have mercy on the poor bastard…" [wildly cackles]

Sources

Amygdalan Arm
The arm of a small Amygdala Great One.
Strictly speaking, the Amygdalan Arm is no trick weapon of
any sort, but certain madmen wield them like clubs.
Starts as a large, tough blunt weapon formed of bone, but
when extended, the hand quivers as if it were still alive.

Tonsil Stone
A latticed, deformed rock, or perhaps a meteorite.
Appears useless, but possesses some odd gravitational force
that prevents its riddance. A dubious soul once said:
"Step lightly round to the right of the great cathedral, and
seek an ancient, shrouded church… The gift of the godhead
will grant you strength…"

Patches dialogue:Confront Patches face-to-face in Lecture Building 1F
"Huh! Oh, this cannot be, you cannot be… No, you didn't… Lord
Amygdala… …How did this come to pass? Hmph! Now, wait just a
moment… Do you think ill of me, me?"

No
"Indeed, I should think not! I shared with you a thing most secret. Now
you're witness to a miracle, and all the stronger for it! You should
appreciate it, if you've a grain of gratitude in you! Heh heh… yes,
appreciate it! Hah hah hah!"

Yes
"Oh, it pains me to hear it. You've made yourself a misreckoning. I shared
with you a thing most secret. Now you're witness to a miracle, and all the
stronger for it! It's plain as a pikestaff.
Now, say in my heart, you were as a lamb to my god. Well, you weren't to
know, and it wasn't for you to know. All's well that ends well, I say. Heh heh
heh!"

"Ahh, well met. My apologies, but I feel a profound thought occuring… On
the good grace of a certain god, and the way he meted out his love. Heh
heh heh…"

"Ahh, well met. This is a most pleasing encounter. You see, I must depart
erelong. My god is lost to me, so I must away to find another.
If the fates are kind, out paths may cross again. Sooner still, if you were
to take the leap… Heh heh heh… Hah hah hah!"